Filling curves for $\mathbb{P}^1 \times \mathbb{P}^1$

TL;DR

This paper determines the minimal bi-degrees of irreducible filling curves over finite fields for the product of projective lines, providing explicit values depending on the size of the field.

Contribution

It explicitly computes the minimal bi-degrees of filling curves on ^1 ^1 over finite fields, filling a gap in the understanding of such curves.

Findings

Minimal bi-degree is (q+1, q+1) for q  2

For q=2, minimal bi-degrees are (4,3) and (3,4)

Provides explicit characterization of filling curves over finite fields.

Abstract

We determine the minimal bi-degree(s) of an irreducible filling curve over $\mathbb{F}_q$ for $\mathbb{P}^1\times \mathbb{P}^1$. It is $(q+1, q+1)$ if $q\neq 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$.